Editing Jordan normal form (section)

== Matrices with entries in a field ==

Jordan reduction can be extended to any square matrix ''M'' whose entries lie in a [[field (mathematics)|field]] ''K''.  The result states that any ''M'' can be written as a sum ''D'' + ''N'' where ''D'' is [[semisimple operator|semisimple]], ''N'' is [[nilpotent matrix|nilpotent]], and ''DN'' = ''ND''. This is called the [[Jordan–Chevalley decomposition]]. Whenever ''K'' contains the eigenvalues of ''M'', in particular when ''K'' is [[algebraically closed]], the normal form can be expressed explicitly as the [[direct sum]] of Jordan blocks.

Similar to the case when ''K'' is the complex numbers, knowing the dimensions of the kernels of {{math|(''M'' &minus; ''λI'')<sup>''k''</sup>}} for 1 ≤ ''k'' ≤ ''m'', where ''m'' is the [[algebraic multiplicity]] of the eigenvalue ''λ'', allows one to determine the Jordan form of ''M''. We may view the underlying vector space ''V'' as a ''K''[''x'']-[[module (mathematics)|module]] by regarding the action of ''x'' on ''V'' as application of ''M'' and extending by ''K''-linearity. Then the polynomials {{math|(''x'' − ''λ'')<sup>''k''</sup>}} are the elementary divisors of ''M'', and the Jordan normal form is concerned with representing ''M'' in terms of blocks associated to the elementary divisors.

The proof of the Jordan normal form is usually carried out as an application to the [[ring (mathematics)|ring]] ''K''[''x''] of the [[structure theorem for finitely generated modules over a principal ideal domain]], of which it is a corollary.